Finite Group Databasebeta

← #85413

Subgroups — #85413

Order 384.

207 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2#5427C211yesmaximal, characteristic, complemented
3#5427C211yesmaximal, characteristic, complemented
4#5426C211yesmaximal, characteristic, complemented
5#5424C211yesmaximal, characteristic, complemented
6#5394C211yesmaximal, characteristic, complemented
7#5389C211yesmaximal, characteristic, complemented
8#3309—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×#41⋊C3C2211yescharacteristic
11C2×GL(2,𝔽3)C2211yescharacteristic
12#898—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#553—31no—
18#550D311yescharacteristic, complemented
19#490—31no—
20#482—31no—
21#480—31no—
22#479—31no—
23#469—31no—
24#466—31no—
25#459—31no—
26#449—31no—
27#449—31no—
28#440—31no—
29#435—31no—
30#417—31no—
31#416—31no—
32#41⋊C3—21no—
33#41⋊C3—21no—
34C2×SL(2,𝔽3)C2311yescharacteristic
35GL(2,𝔽3)—21no—
36C3×C4⋊C4—41no—
37(C2×D3)⋊C4—41no—
38(C2×D3)⋊C4—41no—
39(C2×D3)⋊C4—41no—
40C4⋊Dic3—41no—
41Dic3⋊C4—41no—
42C4×Dic3—41no—
43C2×#41C2×D311yescharacteristic
44C2×#41C2×D311yescharacteristic
45C2×SD16—31no—
46C2×D8—31no—
47C2×D8—31no—
48C2×M16—31no—
49C22×C8—31no—
50#127—31no—
51#126—61no—
52#125—31no—
53#123—61no—
54#121—31no—
55#121—31no—
56#121—31no—
57#121—31no—
58#121—61no—
59C4×Q8C2×D311yescharacteristic
60C4×D4—31no—
61C4×D4—31no—
62C4×D4—61no—
63#117—31no—
64C2×C4⋊C4—31no—
65#107—31no—
66#107—31no—
67#106—31no—
68C4⋊C8—31no—
69#103—31no—
70#103—31no—
71#103—31no—
72#102—31no—
73#102—31no—
74#102—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78#102—31no—
79C22⋊C8—31no—
80C22⋊C8—31no—
81#97—31no—
82C22×D3—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89SL(2,𝔽3)#4111yescharacteristic
90#41—21no—
91#41—21no—
92#41—61no—
93#41—61no—
94C2×Q8C22×D311yescharacteristic
95C2×D4—31no—
96C2×D4—31no—
97C2×D4—31no—
98C2×D4—31no—
99C2×D4—61no—
100C2×D4—61no—
101C2×D4—61no—
102C22×C4—31no—
103C22×C4—31no—
104C22×C4—61no—
105C22×C4—61no—
106SD16—61no—
107SD16—61no—
108D8—121no—
109D8—121no—
110M16—121no—
111C2×C8—31no—
112C2×C8—31no—
113C2×C8—31no—
114C2×C8—31no—
115C2×C8—121no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4S411yescharacteristic
124C22⋊C4—31no—
125C22⋊C4—31no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C42—31no—
131C42—61no—
132C22×C3—41no—
133C2×D3—81no—
134C2×D3—81no—
135C2×D3—81no—
136C3×C4—81no—
137C3×C4—81no—
138C3×C4—81no—
139Dic3—81no—
140Dic3—81no—
141Dic3—81no—
142C23—31no—
143C23—31no—
144C23—61no—
145Q8—31no—
146Q8#23711yescharacteristic
147D4—61no—
148D4—61no—
149D4—61no—
150D4—61no—
151D4—61no—
152D4—61no—
153D4—121no—
154D4—122no—
155D4—241no—
156D4—241no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—241no—
167C2×C4—241no—
168C2×C4C2×S411yescharacteristic
169C2×C4C2×S411yescharacteristic
170C2×C4C2×S411yescharacteristic
171C8—61no—
172C8—61no—
173C8—61no—
174C8—61no—
175C2×C3—41no—
176C2×C3—41no—
177C2×C3—41no—
178D3—161no—
179C22—61no—
180C22—61no—
181C22—61no—
182C22—61no—
183C22—61no—
184C22—61no—
185C22—121no—
186C22—121no—
187C22—121no—
188C22C22×S411yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—31no—
193C4—31no—
194C4—61no—
195C4—121no—
196C4—121no—
197C4—121no—
198C3—41noSylow
199C2—121no—
200C2—121no—
201C2—241no—
202C2#591411yescharacteristic
203C2#592311yescharacteristic
204C2#592511yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#550D3
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C4⋊(#41⋊C3)C2semidirect
2. #5427C2semidirect
3. #5427C2semidirect
4. #5426C2semidirect
5. #5424C2semidirect
6. #5394C2semidirect
7. #5389C2semidirect
18. #550D3semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4⋊(#41⋊C3)2
#54272
#54262
#54242
#53942
#53892
#33093
#8984